Multiplying Exponents

When two terms with exponents are multiplied, it is called multiplying exponents. The multiplication of exponents involves certain rules depending upon the base and the power. Sometimes we need to multiply negative exponents, or multiply exponents with the same base, or different bases. In all these cases, we follow different rules. Let us learn more about multiplying exponents in this article.

Before exploring the concept of multiplying exponents, let us recall the meaning of exponents. An exponent can be defined as the number of times a quantity is multiplied by itself. For example, when 2 is multiplied thrice by itself, it is expressed as 2 × 2 × 2 = 2³. Here, 2 is the base, and 3 is the power or exponent. It is read as '2 raised to the power of 3'.

Now, let us discuss what multiplying exponents mean. When any two terms with exponents are multiplied, it is called multiplying exponents. Let us go through the different cases with the help of examples to understand the concept better.

Consider two terms with the same base, that is, aⁿ and a^m. Here, the base is 'a'. When the terms with the same base are multiplied, the powers are added, i.e., a^m × aⁿ = a^{m+n}

Let us explore some examples to understand how the powers are added.

Example 1: Multiply 2⁴ × 2²

Solution: Here, the base is the same, that is, 2. According to the rule, we will add the powers, 2⁴ × 2² = 2⁽⁴⁺²⁾ = 2⁶ = 64.

Let us verify the answer. 2⁴ × 2² = (2 × 2 × 2 × 2) × (2 × 2) = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶ = 64

Example 2: Find the product of 10⁴⁵ and 10³⁹

Solution: In the given question, the base is the same, that is, 10. According to the rule, we will add the powers, 10⁴⁵ × 10³⁹ = 10^(45+39) = 10⁸⁴.

Will the rule still remain the same if the bases are different? Let us see this in the following section.

When two numbers or variables have different bases, we can multiply the expressions by following some basic exponent rules. Here, we have two scenarios as given below.

When the bases are different and the powers are the same.

Consider two expressions with a different base and the same power aⁿ and bⁿ. Here, the bases are a and b and the power is n. When multiplying exponents with different bases and the same powers, the bases are multiplied first. It can be written mathematically as aⁿ × bⁿ = (a × b)ⁿ

Example: Find the product of 5² and 8²

Solution: Here, the bases are different but the powers are the same. So, applying the rule, we will first multiply the bases, that is, 5² × 8² = (5 × 8)² = 40² = 1600

When the bases and powers are different.

Consider two expressions with different bases and powers aⁿ and b^m. Here, the bases are a and b. The powers are n and m. When the expressions with different bases and different powers are multiplied, each expression is evaluated separately and then multiplied. It can be written mathematically as aⁿ × b^m = (aⁿ) × (b^m)

Example: Multiply the expressions: 10³ × 7²

Solution: Here, the bases and the powers are different. Therefore, each term will be solved separately. 10³ × 7² = 1000 × 49 = 49000.

Let us recall the rules for multiplying exponents with the same base and with different bases in the following figure.

Multiplying Negative Exponents

Negative Exponents tell us how many times we need to multiply the reciprocal of the base. In other words, we can convert a negative exponent to a positive one by writing the reciprocal of the given term and then we can solve it like a positive term. For example, 2^-3 can be written as 1/2³. For multiplying negative exponents, we need to follow certain rules that are given in the following table.

Now, let us understand these rules with the help of the following examples.

Example 1: Find the product of 2^-3 and 2^-9

Solution: Here, the base is the same, that is, 2. The powers are negative and different. Thus, 2^-3 × 2^-9 = 2^-(3+9) = 2^-12 = 1/2¹² = 1/4096 ≈ 0.000244

Example 2: Multiply 6^-3 × 3^-3

Solution: Here, the bases are different and the negative powers are the same. Thus, 6^-3 × 3^-3= (6 × 3)^-3 = 18^-3 = 1/18³ = 1/5832 ≈ 0.0001715

Example 3: Multiply 7^-2 × 6^-3

Solution: Here, both the bases and the negative powers are different. Thus, 7^-2 × 6^-3 = 1/7²× 1/6³= 1/(7²× 6³) ≈ 9.45 × 10^-5

Multiplying Exponents with Variables

If the base of a term is a variable, we use the same exponent rules of multiplication that are used for numbers.

When the variable bases are the same, the powers are added.

Example: Find the product of a⁴ and a¹⁰

Solution: The variable base is the same, that is, 'a'. So, we will add the exponents, a⁴ × a¹⁰= a⁴⁺¹⁰ = a¹⁴

When the variable bases are different and the powers are the same, the bases are multiplied first.

Example: Multiply a¹⁷ × b¹⁷

Solution: The variable bases are different and the powers are the same, that is, a¹⁷ × b¹⁷= (a × b)¹⁷ =(ab)¹⁷

When the variable bases and the powers are different, the terms are evaluated separately and then multiplied.

Example: Find the product of x⁸ and y⁹.

Solution: The variable bases and powers are different, that is, x⁸ × y⁹ = x⁸y⁹

Multiplying Exponents with Square Root

In this section, we will explore the multiplication of exponents where the bases have a square root. It should be noted that the exponent rules remain the same if the bases are square roots.

Apart from this, one important point to be remembered is that we can convert radicals to rational exponents and then multiply the given expressions. For example, the square root of a positive number √a can be expressed as a rational exponent in the following way. √a = a^1/2. Now, when we need to rewrite a given exponential term as a rational exponent, we multiply the existing power with 1/2. For example, if we need to rewrite √5³ as a rational exponent, we will first convert the radical √5 to 5^1/2, then we will multiply the power 3 with 1/2 which makes it 3/2. Now, the radical √5³ is converted to a rational exponent and is written as 5^3/2.

Rules for Multiplying Exponents with Square Root

Now, let us use the exponent rules of multiplication that are applicable to expressions in which the bases are square roots.

When the square root bases are the same, the powers are added.

Example: Find the product of (√5)² and (√5)⁷.

Solution: The square root bases are the same. Thus, (√5)² × (√5)⁷ = (√5)²⁺⁷ = (√5)⁹ = (5)^{1/2 × 9} = (5)^9/2

When the square root bases are different and the powers are the same, the bases are multiplied first.

Example: Multiply (√5)³ × (√7)³

Solution: The square root bases are different and the powers are the same. Thus, (√5)³ and (√7)³= (√5 ×√7)³ = [√(5×7)]³= (√35)³ = (35)^3/2

When the square root bases and the powers are different, the exponents are evaluated separately and then multiplied.

Example: Find the product of (√5)³ and (√7)⁴

Solution: The square root bases and the powers are different. Thus, (√5)³ ×(√7)⁴= 11.18 × 49 ≈ 547.82

Rules for Multiplying Exponents with Fractions

If the base of an expression is a fraction that is raised to an exponent, we use the same exponent rules that are used for bases that are whole numbers. Observe the following table to see the different scenarios.

Let us explore some solved examples to understand this better.

Example 1: Find the product of (2/3)² and (15/8)²

Solution: Here, the fractional bases are different but the powers are the same. Thus, applying the rule given above, (2/3)² × (15/8)² = (2/3 × 15/8)² = (5/4)² = 5²/4² = 25/16

Example 2: Multiply (2/3)² × (2/3)⁵

Solution: Here, the fractional bases are the same. (2/3)² × (2/3)⁵ = (2/3)²⁺⁵ = Thus, (2/3)⁷ = 2⁷/3⁷ = 128/2187.

Example 3: Multiply (3/4)² × (2/3)³

Solution: Here, the fractional bases and the powers are different. So, first, we will solve each term separately and then move further. (3/4)² × (2/3)³ = Thus, (3² × 2³)/(4² × 3³) = (9 × 8)/(16 × 27) = 1/6.

How to Multiply Fractional Exponents?

When a term has a fractional power, it is called a fractional exponent. For example, 2^3/5 is a fractional exponent. Let us understand the rules that are applied to multiply fractional exponents with the help of the following table.

Let us understand these rules with the help of the following examples.

Example 1: Multiply 2^1/2 and 2^3/2

Solution: Here, the bases are the same. Thus, 2^1/2 × 2^3/2 = 2^1/2+3/2 = 2^4/2 = 2² = 4

Example 2: Find the product of 2^1/2 and 3^1/2

Solution: Here, the bases are different but the fractional powers are the same. Thus, 2^1/2 × 3^1/2 = (2×3)^1/2 = 6^1/2 = √6

Example 3: Multiply 4^2/3 × 2^1/3

Solution: Here, the bases and the fractional powers are different. Thus, 4^2/3 × 2^1/3 ≈ 2.52×1.26 = 3.1752

Tips on Multiplying Exponents:

• Zero raised to any power (excluding 0) is 0.
• Any number raised to zero power is 1.
• An exponent is a way of expressing repeated multiplication.

☛ Related Topics

• Exponential Equations
• Irrational Exponents
• Exponential Function
• Operations on Exponential Terms

How does Multiplying Exponents Work?

Multiplying exponents means finding the product of two terms that have exponents. Since there are different scenarios like different bases or different powers, there are different exponent rules that are applied to solve them. There are some basic rules given below that are used in almost all the cases.

• When the terms with the same base are multiplied, the powers are added, i.e., a^m × aⁿ = a^(m+n)
• In order to multiply terms with different bases and the same powers, the bases are multiplied first. This can be written mathematically as aⁿ × bⁿ = (a × b)ⁿ
• When the terms with different bases and different powers are multiplied, each term is evaluated separately and then multiplied. It can be written as aⁿ × b^m = (aⁿ) × (b^m)

Can you Multiply Exponents with Different Coefficients?

Yes, expressions with different coefficients can be multiplied. The coefficients are multiplied separately as shown in the example. For example, 3a² × 4a³ = (3 × 4)×(a² × a³) = 12a⁵.

While Multiplying Exponents do you Add the Powers?

When exponents with the same bases are multiplied, the powers are added. For example, 3⁴ × 3⁵ = 3⁽⁴⁺⁵⁾ = 3⁹

How to Multiply Exponents With Different Bases?

In order to multiply exponents with different bases and the same powers, the bases are multiplied and the power is written outside the brackets. aⁿ × bⁿ = (a × b)ⁿ. For example, 2² × 3² = (2 × 3)² = 6² = 36. However, when we multiply exponents with different bases and different powers, each exponent is solved separately and then they are multiplied. aⁿ × b^m = (aⁿ) × (b^m). For example, 2² × 5⁴ = (2)² × (5)⁴ = 4 × 625 = 2500.

What does Multiplying Exponents with the Same Base mean?

Multiplying exponents with the same base means when the bases are the same while the exponents are different. In this case, the base is kept common and the different powers are added, i.e., a^m × aⁿ = a^(m+n). For example, 2³ × 2⁴ = 2^{(3 + 4)}= 2⁷= 128

How do you Multiply Exponents with Parentheses?

When exponents are multiplied with parenthesis, the power outside the parenthesis is multiplied with every power inside the parenthesis. For example, (2a²b³)² = 2² × a^(2×2) × b^(3×2) = 4a⁴b⁶.

What are the Rules for Multiplying Exponents?

There are different rules that are used in multiplying exponents. The basic rules for multiplying exponents are given below.

• When the expressions with the same base are multiplied, the powers are added, i.e., a^m × aⁿ = a^(m+n)
• When the expressions with different bases and the same powers are multiplied, then the common power is written outside the bracket, i.e., aⁿ × bⁿ = (a × b)ⁿ
• When the expressions with different bases and different powers are multiplied, each term is evaluated separately and then multiplied, i.e., aⁿ × b^m = (aⁿ) × (b^m)

How to Multiply Exponents with Negative Powers?

Multiplying exponents with negative powers follows the same set of rules as multiplying exponents with positive powers. The only difference here is that we should be careful with the addition and subtraction of integers for it. For example, 2^-3 × 2^-9 = 2^-(3+9) = 2^-12 = 1/2¹² = 1/4096 ≈ 0.000244

How to Multiply Exponents with Variables?

In order to multiply exponents with variables, we use the same rules that are used for numbers. For example, let us multiply y⁵ × y³. According to the exponent rule for multiplication with the same base, we add the powers. This means it will be y⁵ × y³ = y^{5 + 3} = y⁸.